Pb1−xCaxTiO3 solid solution (x=0.0, 0.25, 0.50, and 0.75): a theoretical and experimental approach

Pb

Ca

TiO

solid solution (x = 0.0, 0.25, 0.50, and 0.75):

A theoretical and experimental approach

Sérgio Ricardo de Lázaro*

Departamento de Química, Universidade Federal de São Carlos, Caixa Postal 676, 13565-905 São Carlos, SP, Brazil

Poty Rodrigues de Lucena

Instituto de Ciências Ambientais e Desenvolvimento Sustentável, Universidade Federal da Bahia, Barreiras, Ba, Brazil

Júlio Ricardo Sambrano

Laboratório de Simulação Molecular, DM, Universidade Estadual Paulista, Caixa Postal 473, 17033-360, Bauru, SP, Brazil

Paulo Sérgio Pizani

Departamento de Física, Universidade Federal de São Carlos, Caixa Postal 676, 13565-905 São Carlos, SP, Brazil

Armando Beltrán

Departament de Ciències Experimentals, Universitat Jaume I, P.O. Box 6029 AP, 12080 Castelló, Spain

José Arana Varela and Elson Longo

Instituto de Química, UNESP, P.O. Box 355, 14801-970, Araraquara, SP, Brazil

共Received 19 July 2006; revised manuscript received 13 February 2007; published 25 April 2007兲 In this paper we report an experimental and theoretical study based on a periodic density functional inves-tigation into selected compositions of Pb1−xCaxTiO3共x=0.0, 0.25, 0.50, and 0.75兲. Based on our findings, we

propose that the pseudocubic structure of these perovskites presents a long-range tendency for cubic symmetry, while the short-range displacements bring the solid solution to a tetragonal symmetry. The results are discussed in terms of x-ray diffraction, structural optimized parameters, Raman spectroscopy, band structure, density of states, Mulliken charge, and overlap population.

DOI:10.1103/PhysRevB.75.144111 PACS number共s兲: 71.15.Mb, 77.80.Bh, 61.50.Ah, 61.10.Nz

I. INTRODUCTION

ABO₃perovskite oxides constitute an important family of ferroelectrics 共FEs兲 whose relatively simple chemical and crystallographic structures have contributed significantly to our understanding of FE and antiferroelectric phenomena.1 Some prototypical soft phonon mode systems in either pure or mixed crystal form readily undergo structural phase tran-sitions involving both polar and nonpolar distortions of the cubic or tetragonal lattice.2 _PbTiO

3 共PT兲 has attracted con-siderable attention in recent years, because it constitutes one of the major classes of ferroelectric oxides and is an ideal ferroelectric model to study electronic properties in the light of theoretical and experimental results.3,4

The off-center motions of the titanium atom in the center of a tetragonal cell in PbTiO3 are responsible for the spon-taneous polarization of ferroelectric domains by the action of a directional electric field.2 In the ferroelectric tetragonal phase, the Ti atom is displaced in relation to its oxygen cage along the 共001兲 direction.5 _{In perovskite oxides, the} fre-quency shift of the E共1TO兲 soft mode can be correlated to tetragonal-cubic phase transition and the presence of forbid-den Raman peaks to short-range disorder effects. On the other hand, x-ray diffraction 共XRD兲 experiments reveal the long-range arrangement of atoms in a periodic structure.6 Detailed knowledge of the structure in this class of oxides is very important to elucidate the nature of their ferroelectric properties.

Ca-modified lead titanate solid solution, Pb1−xCaxTiO3, has received considerable attention both in thin film and bulk form due to its strong piezoelectric effects and good ferro-electric and pyroferro-electric properties.7–9 _{In the Pb}

1−xCaxTiO3 perovskite structure共PCT兲, the Pb and Ca share the vertices of the unit cell and the titanium is at the center of the struc-ture, surrounded by six oxygen atoms that occupy the middle of the faces in a regular octahedral configuration. Some pa-pers report that calcium doping of the PbTiO3 structure de-creases the tetragonality factor共c/a is the lattice parameters ratio兲.10 _{At around 40% of Ca}2+ _{doping, a phase transition} occurs from the tetragonal to the cubic.

In an early application of x-ray diffraction to the analysis of crystal structure, Vergard observed that many ionic salts alloy a linear relation, at constant temperature, between the crystal lattice constants and concentration.11,12 _{This law has} several physical factors affecting the crystal structures can be readily identified.13 _{These include 共i兲 the relative atomic} sizes of the elements, 共ii兲 the relative volume per valence electron in crystals of the pure elements,共iii兲 Brillouin-zone effects, and共iv兲 electrochemical differences between the el-ements.

Dielectric and Raman experiments performed by Pontes

et al.14_{on PCT thin films as a function of temperature show} a broad ferroelectric phase and a linear dependence of the transition temperature on increasing Ca content. Recently, Chopra and co-workers15 _{reported that PCT is formed at} 650 ° C with a tetragonal structure and that the tetragonality

factor共c/a兲 decreases with increasing concentrations of Ca. Results described in the literature show that the best

ferro-electric properties are around x = 0.25 in

Pb1−xCaxTiO3.8,9,15,16 Correlations between theoretical and experimental techniques can clarify this structural phase transition.

Theoretical-computational studies yield useful informa-tion regarding the electronic and structural properties of sol-ids, which can provide a framework for the interpretation of experimental data in order to determine the influence of dop-ing that often controls important aspects of solid-state chem-istry. Periodic density functional theory 共DFT兲 calculations provide detailed information regarding the properties of solid systems and serve as a fundamental tool for the interpretation of experimental data. Calculations based on the pseudopo-tential method with local-density approximation were corre-lated with results of XANES to elucidate hybridization among atoms in PCT and to identify their density of states 共DOS兲 contributions.17 _{Fuks and co-workers developed a}

thermodynamic formalism based on the

theoretical-computational approach of BacSr共1−c兲TiO3 perovskite solid solution.18

Although ferroelectric and piezoelectric properties of per-ovskites have been extensively studied,19_{few chemical bulk} and surface properties of this material are described in the literature. Crystalline perovskite-type oxides show useful properties and are widely used as piezoelectric and

ferroelec-tric materials. A prime example is the family of

Pb_1−xCaxTiO3 oxides, in particular, PbTiO3.20–22

Here is reported a theoretical and experimental study in which electronic properties are correlated with XRD and Ra-man results in relation to the phase transition induced by the increase in calcium concentration in Pb_1−xCaxTiO3solid so-lution, with x = 0.00, 0.25, 0.50, and 0.75.

This paper is organized as follows. Section II describes the experimental methods, spectral measurement techniques, the computing method, and model systems. Section III dis-cusses the correlation between structural properties of XRD and DFT calculations, experimental Raman results and com-putational details. Section IV summarizes the main lines of our work.

II. METHODOLOGY A. Experimental methods

Pb_1−xCaxTiO3powders were synthesized by the polymeric precursor method 共PPM兲.23–27 In a transparent titanium

cit-rate solution prepared from titanium isopropoxide precursor 共Riedel–de–Haën兲, stoichiometrically proportional amounts of calcium 共first兲 and lead 共second兲 acetate were added slowly to the solution under constant stirring and moderate heating until their complete dissolution. Ammonium hydrox-ide was used to prevent precipitation of lead citrate by ad-justing the pH of the cations solution to around 6.0. In this homogeneous solution, ethylene glycol was added to pro-mote the polyesterification reaction at 90 ° C.

After the water evaporated, the resin was heat-treated at 320 ° C for 2 hours in a static atmosphere, which led to the partial decomposition of the polymeric gel, forming an ex-panded resin composed of partially pyrolized material. The powders were then heat treated in a static air atmosphere at a heating rate of 10 ° C min−1 _{in two heating steps. The first} step was carried out at 350 ° C for 4 hours to pyrolize the organic matter and the second one was carried out at 700 ° C for 2 hours to define the oxide crystalline structure.

Raman measurements were taken with a T-64000 Jobin-Yvon triple-monochromator attached to a charge coupled de-vice共CCD兲 detector. An optical microscope with a 50⫻ ob-jective lens was used to focus the 514.5 nm line of a Coherent Innova 90 argon laser onto the sample.

To evaluate the crystalline evolution and determine the unit cell parameters, we used a Rigaku DMax 2500 PC x-ray diffractometer with Cu K_␣radiation共␭=1.5406 Å兲 operating at room temperature. The data were collected in the 5 ° – 75° 2␪range, with 0.006° step size by 10 s. A metallic Si internal standard was used to correct instrumental errors of the Bragg angle, and a Nelson-Riley extrapolation function was em-ployed to evaluate the experimental lattice constants.6

B. Computing method and model systems

Periodic DFT calculations using Becke’s three parameter hybrid nonlocal exchange functional28 _{combined with the} Lee-Yang-Parr gradient-corrected correlation functional,29 B3LYP, were carried out by means of theCRYSTAL98 com-puter code.30 _{We have been particularly successful in} em-ploying this function in studies of the electronic and struc-tural properties of the bulk and surfaces of PbTiO3共Ref.31兲 and BaTiO₃ 共Ref. 32兲 systems. Pb, Ti, Ca, and O centers

have been described in the scheme关DB兴-31G, 86411-d共41兲, 86-511d3G, and 6-31G*, respectively. Here,关DB兴 stands for the Durand-Barthelat nonrelativistic large effective core TABLE I. Calculated theoretical and experimental lattice parameters共Å兲, tetragonality factor 共c/a兲, and Ti, Oaxial, and Oequatorialatomic

fractional coordinates共x,y,z兲 for Pb_1−xCa_xTiO₃solid solution.

PbTiO3 Pb0.75Ca0.25TiO3 Pb0.50Ca0.50TiO3 Pb0.25Ca0.75TiO3

Theor. Expt. Theor. Expt. Theor. Expt. Theor. Expt.

a = b 3.864 3.913 3.885 3.917 3.864 3.934 3.854 3.947 c 4.153 4.125 3.888 4.015 3.893 3.934 3.858 3.947 c / a 1.075 1.054 1.001 1.025 1.008 1.000 1.003 1.000 Ti 共0.500; 0.500; 0.539兲 共0.500; 0.500; 0.539兲 共0.500; 0.500; 0.500兲 共0.500; 0.500; 0.500兲 Oaxial 共0.500; 0.500; 0.111兲 共0.500; 0.500; 0.103兲 共0.500; 0.500; 0.106兲 共0.50; 0.500; 0.106兲 O_eq. 共0.500; 0.000; 0.625兲 共0.500; 0.000; 0.513兲 共0.500; 0.000; 0.514兲 共0.500; 0.000; 0.514兲

potential.33_{The basis sets can be found at theCRYSTALhome} page.34

As a first step, an optimization procedure of the lattice parameters was carried out, yielding values of a = 3.864 Å and c = 4.153 Å for the PbTiO3共x=0.0兲, space group P4/mm

关see Table I and Fig. 1共a兲兴. Models representing

Pb_1−xCaxTiO3at x = 0.25, 0.50, and 0.75 were built within the supercell 2⫻2⫻1 approach 共Fig.1兲 using tetragonal lattice

constants. A supercell was obtained by defining the new translation vectors in terms of the old vectors. The total en-ergy of all the models were minimized with respect to the tetragonal lattice constants and the z direction of the Ti and O internal atomic coordinates, using the Nelder-Mead minimi-zation method.35

The band structures were obtained at 80 k points along the appropriate high-symmetry paths of the adequate Brillouin zone. Diagrams of the density of states were built to analyze the corresponding electronic structures. The XCRYSDEN

program36 _{was used to draw the band structure diagram.} The theoretical DRX was obtained with the crystallo-graphic software PCW共PowderCell兲 共Ref.37兲 using the

op-timized lattice parameters and internal coordinates.

III. RESULTS

A. Correlation between structural properties of X-ray diffraction and density functional theory calculations

The theoretical and experimental XRD results of the lat-tice constants, atomic fractional coordinates, and tetragonal-ity factor c / a for all models are summarized in TableI.

It is known that different concentrations of calcium in PbTiO3impose structural modifications共decrease of tetrago-nality兲, yielding a structure called pseudocubic. Several ex-perimental works have demonstrated that tetragonality and unit cell volume decrease when calcium is added to the lead titanate lattice.19,38

Our theoretical PbTiO3presents a large tetragonality fac-tor, c / a⫽ 1.075, in good agreement with the XRD experi-mental result, c / a⫽ 1.054, at room temperature. Theoretical values from the PT tetragonal structure 共x=0.0兲 to the pseudocubic structure共x=0.25, 0.50, and 0.75兲 show that the presence of Ca induces slight variations in the cell param-eters, i.e., a decrease of the c parameter, leading to a small decrease of the tetragonality factor共TableI兲 as a function of

the Ca concentration. XRD experimental results reveal the same behavior. Broadly, the theoretical results indicate that the tetragonal lattice tends toward a cubic symmetry. The distorted internal lattice parameters give us an asymmetric result of XRD for the pseudocubic structure. Such variations result from the tendency for progressive modifications in the lattice constants. The literature contains various descriptions of this behavior,8,9,15,19_{which influences the Ca}2+_{ions in the} modification of the lattice constants. The internal fractional coordinates show that the Ti atom is displaced in relation to the symmetric center of the perovskite unit cell. It is impor-tant to observe that, due to the displacement, the Ti atom and its six first neighbors do not form a center with an ideal cubic symmetry, e.g., PbTiO₃ with Pm3m space group.

Similar experimental results were observed by Pontes

et al.19 _{These authors reported an x-ray diffraction 共XRD兲} analysis for several Ca contents. With x = 0.40, the crystal FIG. 1. Pb1−xCaxTiO3cell structure:共a兲 x=0.04; 共b兲 x=0.25; 共c兲

structure appeared as pseudocubic, while Raman scattering revealed a short tetragonality distortion. Some authors asso-ciate the decrease of tetragonality on Pb_1−xCaxTiO3solid so-lution to the increase of the calcium content up to nearly x = 0.40 mole and, as a result of this addition, the perovskite system is arranged in an almost cubic structure 共pseudocu-bic兲.

Analyzing the experimental x-ray diffraction 共Fig. 2兲 of

the powders treated at 700 ° C, it is noted that these oxides are in a structural tetragonal perovskite arrangement. The XRD patterns of the PCT powders reveal some diffraction peak displacements in the PbTiO₃ phase, which reflect the variation in the interplanar distance of the共hkl兲 planes 共001兲 and共100兲, 共110兲 and 共101兲, 共002兲 and 共200兲, 共112兲 and 共211兲, 共202兲 and 共003兲 owing to increased Ca doping.

These diffraction results are indexed according to

tetrag-onal and cubic JCPDS cards Nos. 06-0452 共P4/mm兲 and

43-031 共Pm3m兲, respectively, indicating the formation of perovskite solid solution without the presence of secondary phases, which were undetectable by the XRD technique. The experimental results indicate that, with increasing amounts of calcium, the diffraction profiles show a tendency to form a cubic symmetry.

However, the theoretical XRD pattern of the 0.50 calcium amount in PCT solid solution shows several peaks relating to

a tetragonal structure with a very low relative peak intensity. Figure3 displays selected regions of theoretical diffractions for x = 0.50 and 0.75. The plane reflections presented in the 0.50 diffraction pattern of the theoretical data are not allowed in cubic Pm3m symmetry. Initially, this affirmation is con-firmed by comparing the cubic perovskite ICDD card No. 43-031共Pm3m兲 diffraction with the experimental diffraction pattern in共iii兲 and 共iv兲 of Fig. 2. Thus, for confirmation of the real long-range structure of PCT powders, the linear fit-ting of the interplanar distances to the equations of the cubic and tetragonal cells6_{are compared in TableII. It is observed} that for PCT x = 0.0 and x = 0.25, the XRD profiles are in agreement with the tetragonal cell, and the x = 0.50 and 0.75 proportions show a smaller standard deviation from the cubic adjustment.

In principle, the x-ray technique is an excellent tool for the structural characterization of materials, but its major limitation is revealed in this system: No short range detection 共internal coordinate兲.

In this section, we can make available the Vegard’s law in relation to the atomic size and volume per valence electron in the structure. This effect shows the modification to a long range of the amount of calcium. We observe the decrease of the lattice constant c of linear form for theoretical and ex-perimental results.

TABLE II. Standard deviation共S兲 of lattice parameters a and c of the Pb_1−xCa_xTiO₃solid solution from the interplanar distance.

Tetragonal Cubic

PT PCT0.25 PCT0.50 PCT0.75 PCT0.50 PCT0.75

S_a 0.00435 0.00752 0.02669 0.02199 0.00905 0.00839

Sc 0.00467 0.00922 0.01954 0.01612

FIG. 2. Experimental共up兲 and theoretical 共down兲 x-ray diffrac-tion of Pb1−xCaxTiO3solid system treated at 700 ° C;共i兲 x=0.0, 共ii兲

x = 0.25,共iii兲 x=0.50, and 共iv兲 x=0.75.

FIG. 3. Inset regions of theoretical x-ray diffraction for x = 0.50 and x = 0.75 PCT solid system.

FIG. 4. Raman spectral dependence of the Pb_1−xCa_xTiO₃oxide system.

B. Raman spectroscopy

In a previous statement, the theoretical and experimental results共Table I兲 indicated that the PbTiO3 structure was te-tragonal and the incorporation of calcium decreased the tet-ragonality factor c / a, leading the system to a nearly cubic structure, as demonstrated by the best fit of experimental data and calculated results共TableII兲.

According to Freire et al.,39 _{A1共TO兲 and E共TO兲} vibra-tional modes are associated with dipole moment displace-ments due to the vertical movement of oxygen and titanium in a direction parallel to the c lattice parameter. In tetragonal PbTiO3, each T1umode splits into two vibrational modes in order to form A₁+ E, while the T_2u mode splits into two modes in order to form B1+ E. All Raman and infrared modes are active in this symmetry group. However, in the cubic PbTiO3 with O1h symmetry, the 12 optical-phonon modes transform into the 3T1u+ T2u irreducible representa-tion. The T_1umodes are infrared active and the T_2umode is silent, neither infrared nor Raman active. The T1umodes are doubly degenerated in T1u共TO兲 and T1u共LO兲 modes 共TO and LO are transverse and longitudinal to the k vector of polar-izability, respectively兲.40

The E共1TO兲 mode frequency, the soft mode, is the most sensitive to the decrease in the tetragonality factor of PCT, as

indicated in Fig. 4. However, for x = 0.50 and x = 0.75, the region where PCT would be cubic, where the Raman selec-tion rules predict non-Raman active modes, the spectra clearly show several peaks. The presence of E共1LO兲,

E共2TO兲, E共2LO兲+A共2LO兲, E共3TO兲 peaks can be attributed

to local short-range disorder. An ideal cubic ABO3 perov-skite structure should not display any Raman-active modes in the first-order vibrational spectrum. However, lowering the symmetry leads to changes in vibrational spectra and selection rules, so that first-order Raman scattering is then allowed.41–43

From the Raman spectra共Fig.4兲, it is clear that the

struc-ture does not belong to a tetragonal-cubic transition, but to an increasing symmetry with the conservation of a short-range disorder.

C. Band structures and density of states (DOS)

Figure5 shows the band structure for x = 0.0, 0.25, 0.50, and 0.75. There is a slight variation in the band gap with the increase in Ca concentration. For x = 0.0共PT兲, our theoretical results indicate an indirect band gap 关Fig. 5共a兲兴, 3.23 eV, between X and⌫ points.44,45

When x = 0.25, the top of the valence band is displaced from the X point of the Brillouin zone to the⌫ point. Thus, FIG. 5. Band structure of Pb_1−xCa_xTiO₃for共a兲 x=0.0, 共b兲 x=0.25, 共c兲 x=0.50, and 共d兲 x=0.75.

the optical band gap, 3.13 eV关Fig.5共b兲兴, becomes direct at

the⌫ point. The lowest conduction band 共CB兲 is flat between the⌫-X and Z-⌫ points. The shape between Z and ⌫ points is very similar to the PT case. For x = 0.50关Fig.5共c兲兴, the band gap is direct, 3.03 eV, at the ⌫ point. The CB profile is similar to x = 0.25. When x = 0.75关Fig.5共d兲兴, the top of va-lence band共VB兲 is located at the Z point, and the gap again becomes indirect, 3.85 eV, between the Z and⌫ points. The top of the upper valence band at the ⌫, X, and R points comes out around 0.1 eV below the highest state at the Z point. Again, the lowest CB between⌫-X and Z-⌫ points are flat. For x = 0, 0.25, and 0.50 there is a lower variation in the gap of the oxides; however, at x = 0.75 there is an increase of this value due to the increase of the amount of Ca. This behavior can be a correlation to the existence of a particular molecular structure where the bands are reorganized in rela-tion to PbTiO3bands, but the gap is not significantly modi-fied. The band structures demonstrated the Brillouin-zone ef-fects of Vegard’s law. There is a correspondence among bands and the increase of the amount of calcium. It was observed that at the top of the VB there is a change of X point共PbTiO3兲 and ⌫ and Z points 共PCT兲; this is the influ-ence on the short range of the Vegard’s law.

The density of states in different calcium contents is sum-marized in Fig. 6. The major atomic orbital 共AO兲 of the selected bands was analyzed with a threshold of 0.15 for the eigenvector coefficients.

In calcium concentration of x = 0.25, 0.50, and 0.75, it can be observed that Ca atoms do not contribute in the −5 eV to 3.5 eV energy range; however, this contribution is located in another energy range.17 _{For PbTiO}

3关Fig.6共a兲兴, the analysis of the atomic orbitals共AOs兲 contributions to the top of VB shows mainly derived 2px and 2py orbitals of O atoms. PbTiO3CB is constituted mainly of Ti 3d orbitals.

For Pb_0.75Ca_0.25TiO₃, the VB is constituted of the 2p states of O atoms and display the lowest contributions of the Ti 3dyz state at nearly −4.5 eV 关Fig. 6共b兲兴. The Pb 7s, 6s, and 7p levels are found between −3.5 and −4.5 eV. The main contribution of the CB in this calcium concentration comes from the Ti 3d character.

The calcium content of x = 0.50 mol关Fig.6共c兲兴 shows the same behavior as x = 0.25, with a lower decrease in the Oeq 共see Fig.1兲 2p states character. When x=0.75, the

contribu-tions of Oaxial and Oeq.2p states are displaced to Fermi en-ergy value关Fig.6共d兲兴. The Pb states present the same behav-ior. The main contribution to the CB of Pb0.25Ca0.75TiO3 FIG. 6. Density of states profiles for Ca, Pb, Ti, Oaxialand Oeq.atoms in the Pb1−xCaxTiO3solid solution.

solid solution again comes from Ti 3d levels.

Overall, Pb atoms do not play a preponderant role in the modification of states with the increase of calcium concen-tration in the solid solution. The Ca AOs do not contribute directly to the gap energy range; however, the shape of O, Ti, and Pb AOs are influenced by the calcium substitution. The contributions of Ti and O atoms between −2.5 and −5.0 eV are characterized as hybridizations between the 3d and 2p AOs of these atoms.46 _{For the oxygen atoms, the excessive} contributions in the VB are interactions mentioned as non-bonding states where the electrons are located. For the Ti atoms, the excessive contributions to the CB are nonbonding states where the holes are located.

With the presence of Ca atoms in solid solution, titanium 3d states oriented in the z direction are redistributed to lower energy states.

The DOS results indicates that even though calcium states do not contribute to the band gap and the Pb states DOS profiles hardly change at all, the Ti 3d and O 2p states in Pb1−xCaxTiO3 are perturbed with increased calcium concen-trations. This perturbation of Ti 3d and O 2p states is re-sponsible for the phase transition and increased long-range structural symmetry.

D. Mulliken charges

The atomic charges obtained from Mulliken analysis of all the models are presented in TableIII. The choice of the Mulliken partition is arbitrary, since there is no single method for partitioning the charge density. However, the choice of the given scheme is still extremely useful for com-paring the tendencies revealed in the results of the calcula-tions using similar models.47_{As can be seen in this table, an} increase in the amount of Ca increased the positive charge of the Pb atoms. However, the charge of the Ti atom was not significantly modified. The Oaxial共see Fig. 1兲 showed an in-crease in the negative charge with increasing amounts of calcium. This behavior is due to fact that the O_axialis respon-sible for the simultaneous interactions between Pb-O and Ca-O. This indicates that, as the amount of calcium in-creases, so does the positive charge of the Pb atoms and the negative charge in the Oaxialatoms. The difference in charges among the oxygen atoms 共Table III兲—OeqA. O_eqB, and Oaxial—may be responsible for the distinct ferroelectric and piezoelectric properties due to the influence on the Ca and Pb

atoms when x = 0.25 and x = 0.75. In such cases, the negative charge in both atoms increases, indicating that these atoms are not equivalent to those in the proportions of x = 0.0 and 0.50. Therefore, the number of different neighbors, Pb2+_and Ca2+ ions, yield the formation of an asymmetric center of charges among the various proportions. This asymmetric center of charges occurs because of unequal electron density due to different bonds yielding a heterogeneous interaction field. Such an effect causes an irregularity in the lattice con-stants and the internal coordinate.

The calculations of the electronic structure also give us access to the charge repartition inside the unit cell, for in-stance the TiO₆ cluster. These cluster charges show the in-fluence of the neighboring atoms, Pb and Ca, in relation to the Ti atom. An increase of negative charge in the TiO₆ clus-ter is observed when the amount of Ca increases共TableIII兲,

deforming the electronic structure and modifying the Ti-O interaction. This interaction is very important for the piezo-electric and ferropiezo-electric effect.48_.

E. Overlap population

The overlap population is a measure of the shared elec-tronic density between two atomic nucleus. This result ex-presses that the kind of interaction between atoms conforms to modifications in the crystalline structure. With the crease of overlap population, the covalent interaction de-creases, showing an ionic bond nature. However, the increase of overlap population suggests the improvement of electronic density sharing between atoms, enhancing the covalent inter-action on the oxide bonds. The total overlap population for any pair of atoms is in general made up of positive共bonding兲 and negative共antibonding兲 contributions.49_{A simple way to} know the bonds characteristics in ceramic oxides, is analyz-ing the cation-oxygen interaction. The overlap population re-sults as a function of different calcium concentrations in PbTiO3are presented in TableIV.

Note that the overlap population in the Ti-O and Pb-O bonds decreases and the Ca-O have slight variation. The in-fluence of the quantity of Ca is localized in the displacement of the electronic density from Ti-O and Pb-O bonds. In other words, the Pb for Ca substitution in PbTiO3 host lattice in-creases the ionic nature of PCT solid solution.

The relationship of electronegativity between Pb and Ca atoms in these results is evident. Electronegativity of these elements is evidenced by the difference in the overlap popu-lation of the Pb atom on the undoped and doped PbTiO3host lattice.

IV. SUMMARY

Theoretical and experimental studies of the solid solution pseudocubic phase of Pb1−xCaxTiO3, a product of the substi-TABLE III. Mulliken charges 共in 兩e兩兲 of the atoms in

Pb_1−xCa_xTiO₃unit cells.

PT PCT0.25 PCT0.50 PCT0.75 Ti 2.175 2.184 2.177 2.171 Pb 1.021 1.065 1.063 1.072 Ca 1.593 1.595 1.587 O_axial −1.002 −1.123 −1.172 −1.206 Oeq._a −1.097 −1.087 −1.167 −1.170 O_eq. b −1.169 −1.250 TiO6 −0.520 −0.637 −0.770 −0.846

TABLE IV. Overlap populations共in m兩e兩兲 to Ti-O, Pb-O, and Ca-O bonds.

PT PCT0.25 PCT0.50 PCT0.75

Ti-O 62 58 56 54

Pb-O 20 16 8 9

tution of Pb2+ _{ions for Ca}2+_{ions in the PbTiO}

3 lattice, were conducted employing both XRD and Raman results and pe-riodic first-principles calculations based on the B3LYP func-tional. The main conclusions of this work can be summarized as follows:

共i兲 The bulk equilibrium geometries are in agreement with the experimental results. The structures have a tendency to change from the tetragonal group to the cubic group, whereas the internal coordinates of the Ti atom and its first six neighbors do not form an ideal symmetrical cubic ar-rangement. The XRD and Raman results corroborate the the-oretical analysis. The pseudocubic phase shows characteris-tics of tetragonal and cubic groups.

共ii兲 The calculated optical band gaps are 3.23, 3.13, 3.03, and 3.85 eV for x = 0.0, 0.25, 0.50, and 0.75, respectively. Splitting of the Fermi energy occurs, and the gap is indirect for x = 0.0 and 0.75 and direct for x = 0.25 and 0.50. The z direction of the Ti 3d states stabilizes with the presence of Ca.

共iii兲 Pb+2 _{and Ca}+2 _{ions modify the Mulliken charges of} Oax.and Oeq.. This effect influences the lattice constants and

internal coordinates. These results suggest that the quantity of Ca in PT for x = 0.25 and x = 0.75 modifies the material’s polarizability, which is responsible for the ferroelectricity.

共iv兲 Increasing the amount of calcium augments the ionic character of the Ti-O, especially in the Pb-O interactions, as indicated by the decrease in the overlap population in the Ti-O and Pb-O bonds.

共v兲 It is possible to verify the Vegard’s law in long range 共XDR兲 and short range 共band structure and overlap popula-tion兲. Such law presents macroscopic and microscopic fac-tors for explanation of the lattice contraction in alloys or solid solutions.

ACKNOWLEDGMENTS

This work was supported by the Brazilian research fund-ing agencies FAPESP, CAPES, CNPq and by the Fundació Caixa Castelló-Bancaixa, Spain. The use of the computer facilities of the Servei d’Informàtica 共Universitat Jaume I兲 and Laboratório de Simulação Molecular, Unesp, Bauru, are also gratefully acknowledged.

*Laboratório Interdisciplinar de Eletroquímica e Cerâmica; url: http://www.liec.ufscar.br/ceramica; Electronic address: [email protected]

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